Pros
1: High Burst Damage build into base values.
2: True Damage in his passive= you can kill tanks easier in the early to mid game than most other mages.
3: Mana costs: All it takes to solve your small mana problem is an item that gives 100% mana regen, such as Frost Queen's Claim or Morellonomicon. You don't need an item like Rod of Ages or Tear of the Goddess.
4: His W doesn't scale well with AP, so building an early Rabadon's Deathcap has a surprisingly low powerspike.

Cons
1: IMMOBILE: You are at the mercy of most assassins that have gap closers (Zed, Leblanc, Talon, and especially Akali). you also suffer against tanks with strong CC like Maokai or Alistar.
2: Has all skill shots: it is very easy for your opponents to dodge your skills if they already aren't CCed
3: Low utility. If you fall very behind Vel'Koz offers very little to his team.

R > W > Q > E

R= You high damage ultimate that slows your targets hit. If you are low on health and need to back, you can pop this ultimate to not only do damage to the enemy champion, but you can emergency waveclear with this spell at the same time so you can go back to base and not lose much.

W= Your primary form of reliable damage and waveclear. Always max this first as Maxing Q doesn't give Vel as much early pushing power or harass power.

Q= Your highest single hit basic ability, however it will be stopped by the first unit it hits. Far less reliable of a farming and poking tool than W.

E= Your built in form of CC, it basically has the same aiming and landing mechanic as a Nami bubble. This spell on it's own doesn't do much damage, it's more of a piece of utility so I max this spell last.

Frost Queen's Claim: This item is a core item for several reasons. First, you need something to help you set up your combo E> W > R > Q. The spooky ghost turns Vel Koz from a wave control mage into a long range pick/burst champ. Also Vel Koz's W, his primary damage spell, doesn't scale well with AP so the extra early damage given from F.Q. Claim is much appreciated. Also this item gives Vel Koz just the right amount of mana regen and the gold gen the item gives helps Vel'Koz get his other items much faster. There is no downside to buidling this item on Vel Koz.

Liandry's Torment: SOOOOO much damage. To put it simply, this item gives Vel'Koz a tremendous amount of burst damage.

Ionian Boots: Since a typical Vel'Koz build includes both L. Torment and Void Staff you don't have to build Sorceror's Shoes I prefer Ionian Boots in order to get 10% More CDR to get me a level 18 total of 40% CDR (10% from I. Boots + 15% from Scaling Glyphs + 5% from Masteries + 10% from F.Q.Claim).

Void Staff: Gives a respectable amount of AP. gives good magic penetration, and is relatively cheap.

Rabadon's Deathcap: Later on in your build this item helps give you more damage. I don't like to even think of purchasing this item until after my 4 core items. Before I pick up those items, R. Deathcap feels like a poor early powerspike on Vel since his W, his primary offense spell, doesn't scale well with AP.

Rylae's Crystal Scepter: This is my favorite item to build after my 4 core items because it offers Vel'Koz more utility. It adds a slow to his W, which is quite easy to land in teamfights, adds a stronger slow to his Q, and adds a stronger slow on his R (which is huge because it makes it harder for the enemy champions to escape your burst. Combine that with the natural synergy this item has with L. Torment and Vel'Koz's synergy with L.Torment this item is just beautiful to build on Vel'Koz.

Elixir of Sorcery: This is a great item to chug once you have a full item build. I don't feel like buying it early gives Vel'Koz a tremendous advantage when compared to saving up his money for his core items, but this item is an excellent cherry on top for this champion's damage.

Zhonya's Hourglass: If you are against a very mobile assasin such as Zed, Fizz, LeBlanc, Talong, and especially Akali, rush this item. It is the only item that has a remote chance of keeping him alive vs their burst. If I must purchase this item I will give up my R.C.Scepter to make room for it.

Abyssal Mask: This item is a terrible idea on Vel'Koz. It doesn't provide any CDR, Mana regen which are coveted stats. Instead it offers a more aggressive stat when combining the magic resist with the aura. The issue is if you are close enough to make use of this aggressive aura you are playing Vel'Koz. You never want to get in close range on an enemy champion, you always want to be a medium distance away from your enemy. Even as a defensive item, it sets Vel'koz back because he can't spam his W, which he needs to be able to do in order to have wave control, then he gets two fold set behind.

Luden's Tempest: When compared to the utility of F.Q. Claim, the high synergy of L.Torment, the necessity of Void Staff, the utility/ synergy burst of R.C. Scepter, and the late game power of R.Deathcap this item just doesn't have a place in his kit.

Rod of Ages and Tear of the Goddess: Both of these have the same problem: They need to be built first, they take time or effort to fully come on line, and they mess up the timing on Vel'Koz's mid game powerspike (which is a combination of his kit's base values and the early magic penetration built through items). These items just don't work well on Vek'Koz due to bad timing.

1: Pre 6 combo
E > W> Q > W

2: Post 6 Combo
E > W > R> W > Q
Q is the least reliable spell to hit in your combo and you want to use the precious time you gained by knocking them up with your E in order to hit them with your R.